Messenger Routing Scheme with Time Constrains

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Messenger Routing Scheme with Time Constrains
Vehicle Routing Problem

• Presented by Hui Guo

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Our discussion model

• Partitioned Ad hoc networks
• Networks are partitioned into several clusters
• Transmission is out of range

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Partitioned Ad hoc networks

• Question How the data transmission can be
  achieved?
• Network composition
• Cluster gateway
• Collecting message
• Messenger
• Responsible for delivering messages

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Network model

• Considering a specific cluster
• Suppose each cluster is stable for a period of
  time, i.e., viewed as a stationary node

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Network model

• Assumptions
• Source node generates messages comply with
  Poisson distribution, ? arriving rate
• Messengers are identical, i.e., the same
  traveling speed and buffer space All messages
  are of equal size
• Each route begins and ends at the source node
• The destination of each message is randomly
  selected with uniform distribution across the
  network
• Each message have delivering time constrains ?
• Problem with fixed number of messengers in a
  source node, the maximum message arriving rate ?
  can be supported with do not violate time
  constrains for each message.
• Solve queuing theory

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Network model

• Notations
• m number of messengers at source node
• n number of destination nodes
• T(k) traveling time for visiting k number of
  nodes
• E(T) Expectation of traveling time of a
  messenger
• ?(T) Variance of traveling time of a messenger
• ? Time constrains for message arriving
  destination
• s messenger capacity (carrying s messages at
  each time)
• Objective
• Get the maximum supported arriving rate (?) of
  messages

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Some solutions

• Idea
• M messengers viewed as m number of servers that
  providing service in the queuing system
• The traveling time of messenger viewed as service
  time
• The sum of waiting time and service time should
  not more than time constraints ?
• i.e. Twait Tserv ? ?
• Tserv E(T)

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Some solutions

• p(k) probability of visiting k number of
  destinations when carrying s number of messages
• Calculating the distribution of service time

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M/G/1 model

• For one messenger per source node
• Service rate ? s/E(T)
• Pollaczek-Khintchine Formulas
• Then, we have
• to get the value of ?

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G/G/m model

• For multiple messengers per source node
• Service rate ? s/E(T)
• ?x variance of random variable x that represents
  interarrival time of message.
• Suppose arriving messages comply with Poisson
  arrival with expectation ?, then x comply with
  negative exponential distribution
• Then
• We have

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Vehicle Routing Problem (VRP)

• A problem within opertional research
• Problem definition
• N identical vehicles of capacity C
• M customers with traffic demands digt0, i 1..M
• Each vehicle serves subset of customers
• all customers served, each once by one of the
  vehicles
• Vehicles routes start and end at the depot
• Objective
• total cost (distance) is minimal

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Vehicle Routing Problem

• Make a set of vehicles visit a set of locations
• Vehicles have limited capacity
• Each location is visited by a vehicle only once
• Visits have time windows
• Minimize total distance traveled

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Application of VRP

• Repair/install equipment
• Pick up money from banks
• Deliver prisoners from jail to court etc
• Street cleaning, garbage collection
• Automated guided vehicles in a factory
• Ambulance routing
• Drilling circuit boards
• Robot arm movements
• Computer networks

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Traveling Salesman Problem

• a lower-level type of routing problem
• Traveling Salesman Problem (TSP)
• TSP is NP-hard Problem
• (C. Papadimitriou. The Euclidean traveling
  salesman problem is np-complete. Theoret. Comput.
  Sci., 4237.244, 1977)
• (N!)

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Traveling Salesman Problem

• Heuristic algorithms
• Nearest neighbor
• Cheapest Insertion
• Clarke-Wright
• Local optimization algorithms
• 2-opt, 3-opt first proposed by Croes 1958

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Variants of VRP

• Capacitated Vehicle Routing Problem (CVRP)
• Vehicle Routing Problem with Time Windows (VRPTW)
• Multiple Depot Vehicle Routing Problem (MDVRP)
• Vehicle Routing Problem with Pick-up and
  Delivering (VRPPD)
• Vehicle Routing Problem with Backhauls (VRPB)
• And so on

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Algorithms

• Heuristics
• Clarke and Wright
• G. Clarke and J. Wright "Scheduling of vehicles
  from a central depot to a number of delivery
  points 1964.
• Matching Based
• M. Desrochers, and T. W. Verhoog. "A Matching
  Based Savings Algorithm for the Vehicle Routing
  Problem". 1989.
• Multi-Route Improvement
• G. A. P. Kinderwater and M. W. P. Savelsbergh.
  "Vehicle Routing Handling Edge Exchanges". 1997.
  
• 2-Phase Algorithm Petal Sweep
• Other
• Metaheuristics
• Ant System
• B. Bullnheimer, R. F. Hartl and C. Strauss.
  "Applying the Ant System to the Vehicle Routing
  Problem". 1997.
• Deterministic Annealing
• G. Dueck and T. Scheurer. "Threshold Accepting
  A General Purpose Optimization Algorithm". 1990.
• Genetic Algorithms
• S.R. Thangiah. Vehicle Routing with Time Windows
  using Genetic Algorithms, 1993
• Tabu search
• P. Badeau, M. Gendreau, F. Guertin, J.-Y. Potvin
  and E. Taillard. "A Parallel Tabu Search
  Heuristic for the Vehicle Routing Problem with
  Time Windows".1997.